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Viviani Surface and Its Intersection Curve

Viviani Surface and Its Intersection Curve
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2025-10-17 00:43
Coordinate GeometrySolid GeometryTransformationParametric EquationGraphing
In collections intersection: surface-surface, curve-curve
① This example demonstrates the Viviani surface and its intersection curve with a sphere, known as the Viviani curve, a classic three-dimensional solid geometry figure. ② Core mathematical concepts include sphere, cylinder, parametric equations of surfaces and curves, spatial curve construction, and geometric transformations such as rotation and scaling. ③ Construction highlights: from the construction summary, GeoGebra commands such as Point, Sphere, Line, CurveCartesian, Rotate, Dilate, Midpoint, Circle, and Cylinder are used to create spheres, cylinders, auxiliary points and lines, and to generate the Viviani curve and surface via parametric curves (CurveCartesian) and rotation (Rotate), with LaTeX text annotations. ④ Suitable for high school to university level solid geometry, analytic geometry, or mathematical visualization teaching, as well as demonstrations in mathematical modeling or computer-aided geometric design (CAGD).

GeoGebra Commands & Parameters

  1. Point(zAxis) → A
  2. Sphere((0, 0), z(A)) → b
  3. Line(D, xAxis) → f
  4. CurveCartesian((R * (1 + cos(t))) + r - (2 * R), (R * sin(t)), ((2 * sqrt((R * r) - R^(2))) * sin(t / 2)), t, 0, (4 * pi)) → viv
  5. Rotate(viv, ((-pi)) / 2, yAxis) → d'
  6. Dilate(b, 0.99, (0, 0)) → b'
  7. Midpoint(A, D) → B
  8. Line(B, xAxis) → g
  9. Circle(g, A) → k
  10. Cylinder(k, (-(r - 0.05))) → l, l_{2}, l_{3}
  11. Cylinder(k, r - 0.05) → i, i_{2}, i_{3}
  12. LaTeX(viv, true, true) → text1

AI Drawing Prompt Reference

Draw a Viviani surface with its intersection curve against a sphere—the Viviani curve, a classic three-dimensional geometric figure. First create a sphere, then select a point on its surface and construct a cylinder with radius equal to the distance from that point to the sphere's center, so that the cylinder intersects the sphere along the Viviani curve. Use parametric equations to draw this spatial curve, add coordinate axes for reference, and include mathematical labels to complete the solid geometry visualization.
The prompt above was compiled from the original drawing commands and is for reference only.